Scalable algorithms for locally low-rank matrix modeling

We consider the problem of modeling data matrices with locally low-rank (LLR) structure, a generalization of the popular low-rank structure widely used in a variety of real-world application domains ranging from medical imaging to recommendation systems. While LLR modeling has been found to be promising in real-world application domains, limited progress has been made on the design of scalable algorithms for such structures. In this paper, we consider a convex relaxation of LLR structure and propose an efficient algorithm based on dual projected gradient descent (D-PGD) for computing the proximal operator. While the original problem is non-smooth, so that primal (sub)gradient algorithms will be slow, we show that the proposed D-PGD algorithm has geometrical convergence rate. We present several practical ways to further speed up the computations, including acceleration and approximate SVD computations. With experiments on both synthetic and real data from MRI (magnetic resonance imaging) denoising, we illustrate the superior performance of the proposed D-PGD algorithm compared to several baselines.